Abstract: The regular FI-algebras are built up on general sets merely by the implication operation. In this paper, based on axiom groups and internal relations of properties of regular FI-algebras, two characterizations of regular FI-algebras with fewer axioms are given, which simplify the definition of regular FI-algebras. Moreover, implicative distributivity is introduced into regular FI-algebras and some properties of implicative distributive FI-algebras are discussed. It is proved that implicative distributive regular FI-algebras and Boolean algebras are equivalent algebraic structures. A new characterization for Boolean algebras is given, showing them to be more similar to two-valued logic algebras in form.

Key words: FI-algebra, regular FI-algebra, implicative distributivity, Boolean algebra

摘要： 正则FI-代数是仅基于蕴涵算子在一般集合上建立的逻辑代数。基于正则FI-代数的公理组以及诸多性质之间的内部联系，给出了正则FI-代数的两个公理组条件更少的刻画定理，简化了正则FI-代数的定义形式。在正则FI-代数中引入蕴涵分配性，探讨了蕴涵分配正则FI-代数的若干性质，证明了蕴涵分配正则FI-代数与Boole代数是相互等价的代数系统，给出了Boole代数的一种新的刻画，使其在形式上更接近于二值逻辑代数。

关键词: FI-代数, 正则FI-代数, 蕴涵分配性, Boole代数